T-branes and 3d Mirror Symmetry - Caltech Particle...

T-branes and 3d Mirror Symmetry

Roberto Valandro

University of Trieste

Caltech, 22 February 2016

Based on 1602.0XXXX with A. Collinucci, S. Giacomelli and R. Savelli

Roberto Valandro T-branes and 3d Mirror Symmetry

Introduction

F-theory: connection between geometry and gauge theory ( + sugra ):

IIB/IIA gauge theory on D7/D6 ↔ F/M-theory on (singular) T 2/S1 fibrations

Eg, N = 4 D-branes: G = U(4) ↔ u · v = z4 : A3 sing

Vev for complex adj Φ ↔ u · v = det(z 1N − Φ)deform of D-br config deform of sing

Φ =

λ1 0 0 00 λ2 0 00 0 λ3 00 0 0 λ4

↔ u · v =∏4

i=1(z − λi )

D-branes split: G = U(1)4 ↔ No singularity (deformed away)


Introduction

IIB/IIA gauge theory on D7/D6 ↔ F/M-theory on (singular) T 2/S1 fibrations

Eg, N = 4 D-branes: G = U(4) ↔ u · v = z4 : A3 sing

Vev for complex adj Φ ↔ u · v = det(z 1N − Φ)deform of D-br config deform of sing

Φ =

0 1 0 00 0 0 00 0 0 00 0 0 0

↔ u · v = z4

Bound state G = U(1)× U(2) ↔ Geometry untouched

T-brane ↔ ???

[Cecotti, Cordova, Heckman, Vafa; Heckman, Tachikawa, Vafa, Wecht; Donagi, Wijnholt]

Geometry does not see all possible vevs: sensible only to Casimirs


IntroductionT-brane is vev for Φ (strings between D6/D7s). What is M-theory lift?

M2TN1 TN2

∗ Φ11 and Φ22 lift to geometric c.s. deformations,∗ Φ12 lift to M2-brane wrapping the vanishing cycle.

T-brane (vev for Φ12) coherent state of M2-branes on vanishing cycles.

In absence of formulation for microscopic M2, find a framework that allows todescribe this extra data in M-theory. [Anderson, Heckman, Katz; Collinucci, Savelli]

Here we turn to the 3d perspective of a probe M2-brane that see this effect.


Theory ALet’s start by considering a stack of N D6-branes and we probe them by a D2.

0 1 2 3 4 5 6 7 8 9N × D6 × × × × × × ×

D2 × × ×

N = 4 d = 3 U(1) theory with

U(1) NAµ, φ1, φ2, φ3

Q̃I

QI

? One Abelian vector multiplet

Aµ, φ3, ϕ ≡ φ1 + iφ2

? N flavors hypermultiplets

QI , Q̃I with I = 1, ...,N

? SuperpotentialWA = ϕ

∑I

Q̃IQI

? Moduli space separates into two branches.


Theory A - Higgs Branch

U(1) U(N)

Q̃I

QI

Coordinates: gauge invariant mesons. MIJ ≡ QIQ̃J (I = 1, ...,N) s.t.

rk M = 1 and Tr M = 0 (dimC = 2N − 2)

→ It has manifest U(N) global symmetry.

Deformations (lift some directions):

I complex masses δW = mcQ1Q̃1

I real masses∫

d4θQ†1 emr θθ̄Q1

(mc ,mr ) are the vev for the D6-brane scalars: move D6s apart from D2.


Theory A - Coulomb Branch

Photon can be dualised: ∗F = dγ(= JT )

Aµ, φ1, φ2, φ3 ↪→ V± ∼ e±(φ3+iγ), ϕ ≡ φ1 + iφ2

U(1) U(N)V±, ϕ

Coordinates: V± and ϕ s.t.

V+V− = ϕN (dimC = 2)

AN−1 singularity (dual photon↔ M-theory circle)

Deformations:

I mc ⇒ deformations of CB: ϕN 7→ ϕN−1(ϕ+ mc)

I mr ⇒ resolutions of CB.


Theory A

T-brane deformation: (For 4D analog, see [Heckman, Tachikawa, Vafa, Wecht])

〈ΦD6〉 = mT =

0 0 · · · 0m 0 · · · 0...

.... . .

0 0 0

δW = m Q1Q̃2

∗ It breaks U(N) flavor symmetry to U(1)× U(N − 2) and makes twochiral multiplets massive.

∗ What about Coulomb branch? Integrating out massive flavors, obtainN − 2 flavors with couplings ϕQIQ̃I and one with ϕ2

m Q′Q̃′. Hence

V ′+V ′− = ϕN−2 ϕ2

m

AN−1 singularity untouched. [Benini,Closset,Cremonesi]

This description is ok and we could proceed and study T-brane from this p.o.v., butintrinsically perturbative and hard to generalise to more interesting groups.


3d mirror symmetry [ Intriligator, Seiberg ]

Theory A (3d SQED with N flav) dual to another 3d theory, Theory B.

String theory interpretation: TST composition or through ‘9-11’ flip

M2 at C2/ZN × R3 × C2

D2, N × D6’s on R10 D2, D6 on C2/ZN × R6

S1S1

TST

Theory B: completely different theory, but

? on the same singular space of M-theory: form that can be generalized;

? string theory at sing: power of quiver techniques.

⇒ Map T-brane deformations here.


Theory B

U(1)

U(1)

U(1)

q̃1

q1

q̃2

q2

q̃3

q3

ϕ2,W2,±

ϕ1,W1,± ϕ3,W3,±

N = 4 U(1)N gauge theory∗ Each node comes with a vector multiplet dualize photon: (ϕi ,Wi,±)

∗ N hypermultiplets (qi , q̃ i ), connecting the nodes.∗ Superpotential

W =N∑

i=1

Siqi q̃ i (Si ≡ ϕi − ϕi+1)

∗ Again, HB and CB.


Theory B - Higgs Branch

1

2

3

q̃1

q1

q̃2

q2

q̃3

q3

Coordinates: gauge inv comb’s qi q̃i , B ≡ q1q2...qN and B̃ ≡ q̃1q̃2...q̃N

◦ Superpotential W =∑N

i=1 Siqi q̃ i −Ψ∑N

i=1 Si → F-terms:

qi q̃i = Ψ ∀i ⇒ B B̃ = ΨN (AN−1 sing)

DeformationsI complex FI-terms δW = ζ S1 ⇒ q1q̃1 = Ψ− ζ deform.I real FI-terms ξ

∫d4θVi blow-up.


Theory B - Coulomb Branch

U(1)

U(1)

U(1)

q̃1

q1

q̃2

q2

q̃3

q3

ϕ2,W2,±

ϕ1,W1,± ϕ3,W3,±

Coordinates: (ϕi ,Wi,±) s.t.

Wi,+Wi,− = Si Si−1 (Si ≡ ϕi − ϕi+1)

Symmetries> U(1)N−1 topological sym: γi 7→ γi + ci currents: JU(1)i = ∗Fi = dγi .

> Wi,± ∼ e±(ϕi +iγi ) have qi = ±1 in supermultiplets with currents Ji,±.⇒ enhanced global sym: SU(N).


3d Mirror Map [ Intriligator, Seiberg; Aharony, Hanany, Intriligator, Strassler, Seiberg ]

Map between gauge invariant coord defining CBA ↔ HBB and HBA ↔ CBB :

CBA ↔ HBB :

ϕ ↔ Ψ = qi q̃i V+ ↔ B V− ↔ B̃

V+V− = ϕN ↔ B B̃ = ΨN

HBA ↔ CBB :

Mii ↔ Si Mi−1

i ↔ Wi,+ Mii−1 ↔ Wi,−

rk M = 1 Mi−1i−1Mi

i −Mi−1iMi

i−1 ↔ Si−1Si = Wi,+Wi,−

Tr M = 0 ↔∑

i

Si = 0

( SU(N) symmetry )


T-brane

Theory A: off-diagonal mass term +m Q1Q̃2

W = ϕ∑

I

QIQ̃I +∑I,J

(mT )JIQIQ̃I with mk

T = 0

F-terms:(ϕ1 + mT ) ·Q = 0 Q̃ · (ϕ1 + mT ) = 0

Construct gauge invariant F-term:

(ϕ1 + mT ) ·M = 0 M · (ϕ1 + mT ) = 0

Theory B: by mirror map obtain how +m W2,+ should modify F-terms

(Ψ1 + mT ) ·

S1 W2,+

W2,− S2

. . .

= 0

mT breaks SU(N) symmetry


T-brane in theory B

Example N = 3, δW = m W2,+:

1

2

3

q̃1

q1

q̃2

q2

q̃3

q3

ϕ2,W2,±

ϕ1,W1,± ϕ3,W3,±

Ψ 0 0m Ψ 00 0 Ψ

· S1 W2,+ W1,−

W2,− S2 W3,+

W1,+ W3,− S3

= 0

One can apply a bi-unitary transformation to bring into the form:


T-brane in theory B

Example N = 3, δW = m W2,+:

1

2

3

q̃1

q1

q̃2

q2

q̃3

q3

ϕ2,W2,±

ϕ1,W1,± ϕ3,W3,±

m 0 00 Ψ2

m 00 0 Ψ

· > > >

> S′2 W3,+

> W ′3,− S3

= 0


T-brane in theory B

Example N = 3:

1 3

q̃′, q3

q′, q̃3

W ′1,± W ′

3,±

m 0 00 Ψ2

m 00 0 Ψ

· > > >

> S′ W ′3,+> W ′3,− S3

= 0

New F-terms captured by effective superpotential:

Weff = S′q′q̃′ + S3q3q̃3−Ψ2

mS′ + Ψ S3

HBB : q3q̃3 = Ψ q′q̃′ =Ψ2

m⇒ B′B̃′ =

Ψ3

mA2 sing

CBB : W ′3,+W ′3,− = S′S3


Effective quiver and effective superpotential

1

2

3

N 4

S1 q̃1

q1

q̃2 S2

q2

q̃3q3q̃N qN

−−−−→m W2,+

1 3

N 4

S′ q̃′

q′q̃3q3q̃N qN

W =N∑

i=1

Si (qi q̃ i−Ψ) −→ Weff = S′(

q′q̃′ − Ψ2

m

)+

N∑i=3

Si (qi q̃ i−Ψ)

( Check: if add m W2,− = m S′, nodes 1 and 3 fuse togehter and sing becomes AN−3

→ consistent with mirror of diagonalisable mass deform )


Effective quiver: non-minimal T-brane

mT =

0 0 0 · · ·m 0 0 · · ·0 m 0 · · ·...

......

. . .

1 3

N 4

S′ q̃′

q′q̃3 S3q3q̃N qN

−−−−→m W3,+

1

N 4

S′′ q̃′′

q′′q̃N qN

Weff −→ W ′eff = S′′(

q′′q̃′′ − Ψ3

m2

)+

N∑i=4

Si (qi q̃ i −Ψ)

B′′B̃′′ =Ψ3

m2 ΨN−3 ( still AN−1 sing )


SU(4) → SU(2)

mT =

0 0 0 0m 0 0 00 0 0 00 0 m 0

δW = mW2,+ + mW4,+

1

2

3

4

q̃1

q1

q̃2

q2

q̃3q3

q̃4q4

−−−−−−−−−−−−→m W2,+ + m W4,+ 1 3

q̃′, q′′

q′, q̃′′

W =N∑

i=1

Si (qi q̃ i−Ψ) → Weff = S′(

q′q̃′ − Ψ2

m

)+S′′

(q′′q̃′′ − Ψ2

m

)


Obstruction of blow-up by T-brane

Theory A:

∗ Blow-up↔ real mass ( mr ∈ bkgr vector-multiplet )∫d4θ

(Q†i emi

r θθ̄Qi − Q̃ i†emirθθ̄Q̃ i

)∗ δW = m Q1Q̃2 breaks background sym U(1)1−2

corresponding bckgrnd vector mult no more available ⇒ m1r = m2

r

⇒ mass terms with m1r 6= m2

r forbidden (D61 and D62 are bound together).

Theory B:

∗ Blow-up↔ real FI-term∫

d4θ ξb VU(1)2 ∼∫

d4θVb ΣU(1)2 .

∗ δW = m W2,+ breaks topological U(1)2 symmetry

Vb has gone and the above coupling forbidden

⇒ blow-up relative to node 2 is obstructed. [Anderson,Heckman,Katz]


Conclusions

We find the mirror theory for D2-branes probing a stack of T-branes.

I On the A-side, off-diagonal mass deformation. On the B-side, viamonopole operators.

⇒ This provides us with definition of a T-brane directly in terms of amembrane probing a singularity.

I Found effects of T-brane on HB and CB on B-side: HB describingprobed geometry does not change, CB is partially lifted and globalsymmetry broken.

I Effective description of probe theory by Weff and effective quiver.

Open problems:

∗ What about other ADE grops?

∗ Probing more complicated geometries (e.g. with matter curves).

∗ Impact of T-brane on Hilbert series approach to 3d moduli spaces.


Thank you!


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